

A300153


Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the number of parts inscribed in a rose or rhodonea curve with polar coordinates r = cos(t * (k/n)).


2



1, 4, 4, 2, 1, 3, 8, 12, 12, 8, 3, 4, 1, 4, 5, 12, 20, 24, 24, 20, 12, 4, 2, 9, 1, 10, 3, 7, 16, 28, 4, 40, 40, 4, 28, 16, 5, 8, 12, 12, 1, 12, 14, 8, 9, 20, 36, 48, 56, 60, 60, 56, 48, 36, 20, 6, 3, 2, 4, 20, 1, 21, 4, 3, 5, 11, 24, 44, 60, 72, 80, 84, 84, 80
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OFFSET

1,2


COMMENTS

For any real p > 0, the rose or rhodonea curve with polar coordinates r = cos(t * p):
 is dense in the unit disk when p is irrational,
 is closed when p is rational, say p = u/v in reduced form; in that case, the number of parts inscribed in the curve is T(v, u),
 see also the illustration in Links section.


LINKS

Table of n, a(n) for n=1..74.
Rémy Sigrist, Illustration of the first terms
Eric Weisstein's World of Mathematics, Rose
Wikipedia, Rose (mathematics)


FORMULA

T(1, k) = A022998(k).
T(n, k) = T(n/gcd(n, k), k/gcd(n, k)).
Empirically, when gcd(n, k) = 1, we have the following formulas depending on the parity of n and of k:
 k is odd  k is even
++
n is odd  T(n, k) = k * A029578(n+1)  T(n, k) = 2 * k * n
n is even  T(n, k) = 2 * k * A029578(n+1)  N/A


EXAMPLE

Array T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9
+
1 1 4 3 8 5 12 7 16 9
2 4 1 12 4 20 3 28 8 36
3 2 12 1 24 10 4 14 48 3
4 8 4 24 1 40 12 56 4 72
5 3 20 9 40 1 60 21 80 27
6 12 2 4 12 60 1 84 24 12
7 4 28 12 56 20 84 1 112 36
8 16 8 48 4 80 24 112 1 144
9 5 36 2 72 25 12 35 144 1
10 20 3 60 20 4 9 140 40 180
11 6 44 18 88 30 132 42 176 54
...
The following diagram shows the curve for T(2, 1) and the corresponding 4 parts:

######## ########
##### ####### #####
### ### ### ###
### ##  ## ###
## ## ## ##
## # Part #2 # ##
## ## ## ##
# ###  ### #
#   Part #3  #######  Part #1   #
# ###  ### #
## ## ## ##
## # Part #4 # ##
## ## ## ##
### ##  ## ###
### ### ### ###
##### ####### #####
######## ########



CROSSREFS

Cf. A022998, A029578.
Sequence in context: A196766 A153163 A168455 * A182781 A291085 A193556
Adjacent sequences: A300150 A300151 A300152 * A300154 A300155 A300156


KEYWORD

nonn,tabl


AUTHOR

Rémy Sigrist, Feb 26 2018


STATUS

approved



